How to set a specific distance between two vertices?
In this particular case, you can slide (G G)the vertex down to the bottom corner, and then G Z and '0.77' to move it back again.

In general, when things aren't nicely lined up, 


With snap set to 'Vertex' and 'Active',
First create a Custom Orientation (Ctrl AltSpace) from the edge you want to measure down (or parallel to).
ShiftD duplicate any vertex to make one you can use as a snapping source/target.
Snap your new loose vertex to a starting point
GYY+(numerical entry) your vertex parallel to your Custom orientation.


you now have a measured point to snap other things to.

I use a lot of Custom Orientations, so I check 'Use after Creation' and 'Overwrite Previous' in the Tool Shelf.

Using Custom Orientations and a loose vertex, with the pivot set to 'Active Element', you can extend this method to measure angles around an axis in the custom edge's direction, through the active element (RYY+(numerical entry))